This book covers the main topics concerned with interpolation and approximation by polynomials. This subject can be traced back to the precalculus era but has enjoyed most of its growth and development since the end of the nineteenth century and is still a lively and flourishing part of mathematics. In addition to coverage of univariate interpolation and approximation, the text includes material on multivariate interpolation and multivariate numerical integration, a generalization of the Bernstein polynomials that has not previously appeared in book form, and a greater coverage of Peano kernel theory than is found in most textbooks. There are many worked examples and each section ends with a number of carefully selected problems that extend the student's understanding of the text.
George Phillips has lectured and researched in mathematics at the University of St. Andrews, Scotland. His most recent book, Two Millenia of Mathematics: From Archimedes to Gauss (Springer 2000), received enthusiastic reviews in the USA, Britain and Canada. He is well known for his clarity of writing and his many contributions as a researcher in approximation theory.
Author(s): George M. Phillips
Series: CMS Books in Mathematics
Edition: 1
Publisher: Springer
Year: 2003
Language: English
Pages: 327
City: Berlin
Preface......Page 8
Contents......Page 14
1.1 Introduction......Page 16
1.2 The Vandermonde Equations......Page 31
1.3 Forward Differences......Page 43
1.4 Central Differences......Page 55
1.5 q -Differences......Page 58
2.1 The Legendre Polynomials......Page 64
2.2 The Chebyshev Polynomials......Page 79
2.3 Finite Point Sets......Page 97
2.4 Minimax Approximation......Page 102
2.5 The Lebesgue Function......Page 115
2.6 The Modulus of Continuity......Page 131
3.1 Interpolatory Rules......Page 134
3.2 The Euler–Maclaurin Formula......Page 148
3.3 Gaussian Rules......Page 158
4.1 Peano Kernels......Page 162
4.2 Further Properties......Page 168
5.1 Rectangular Regions......Page 178
5.2 Triangular Regions......Page 191
5.3 Integration on the Triangle......Page 203
5.4 Interpolation on the q-Integers......Page 210
6.1 Introduction......Page 230
6.2 B-Splines......Page 233
6.3 Equally Spaced Knots......Page 244
6.4 Knots at the q-Integers......Page 254
7.1 Introduction......Page 262
7.2 The Monotone Operator Theorem......Page 278
7.3 On the q-Integers......Page 282
7.4 Total Positivity......Page 289
7.5 Further Results......Page 295
8.1 The q-Integers......Page 306
8.2 Gaussian Polynomials......Page 311
References......Page 320
D......Page 324
I......Page 325
O......Page 326
W......Page 327
